Download CPSC 311: Analysis of Algorithms Proof by Induction Example

CPSC 311: Analysis of Algorithms
Proof by Induction Example Problems
These problems are solely for your own benet in practicing inductive proofs. They will not be
turned in or graded. A solution sheet will not be handed out. If you want to discuss whether you
are doing them correctly, you can meet with the instructor.
Prove the following statements using induction.
References: A Logical Approach to Discrete Math by David Gries and Fred B. Schneider,
Springer-Verlag, 1994, and The Art of Computer Programming: Vol 1 by Donald Knuth, AddisonWesley, 1973.
For all n 4, there exist nonnegative integers h and k such that 2 h + 5 k = n.
For all n 0, =1i = n (n + 1)=2.
,1 (2 i + 1) = n2.
For all n 0, =0
,1 3 = (3 , 1)=2.
For all n 0, =0
For all n 0, 22 , 1 is divisible by 3.
Dene the value n! recursively for all n 0 as follows.
Let 0! = 1 and let n! = n (n , 1)! for all n > 0. Prove that n! = 1 2 : : : (n , 1) n.
7. Prove that n! > 2 ,1 for all n 3.
8. Dene a sequence of numbers S recursively for all n 0 as follows.
Let S0 = 0. Let S = 2 S ,1 + 1 for all n > 0. Prove that S = 2 , 1 for all n 0.
9. What is wrong with the following `proof'?
Claim: Let a be any positive number. For all positive integers n, we have a ,1 = 1.
Proof:
Basis: If n = 1, a ,1 = a1,1 = a0 = 1.
Inductive Hypothesis: Assume a ,1 = 1 for all n = 1; 2; : : : ; k.
We now show that it is true for n = k + 1:
,1
,1
( +1),1)
a
= a = a a ,a2 = 1 1 1 = 1:
The inductive hypothesis was used to replace a ,1 and a ,2 by 1.
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